There are little lakes (I would call them ponds, because a good thrower could easily throw a baseball across them) and some larger lakes all over the place. I don't know if you're allowed to fish at them, but people are always around them. There are a lot along the undeveloped section of International Drive near Chatham.
I think there is catch and release fishing at Mickey's Retreat which is across the street from Chatham, but I'm not positive.
I'm sure you could find somewhere to go...Florida has billions of "lakes"